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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"becddb14303f8f51d73c29c4951d3087a129967f3344574f531d0c72b78a1f4d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-2eda446fe9bdb6ec","role":"sequence","name":"E2(n, k), the second-order Eulerian numbers with E2(0, k) = \u03b4_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,    1;\n  2: 1,    5,      2;\n  3: 1,   17,     26,       6;\n  4: 1,   45,    196,     154,       24;\n  5: 1,  105,   1036,    2104,     1044,      120;\n  6: 1,  229,   4470,   19052,    22940,     8028,      720;\n  7: 1,  481,  17070,  134090,   322820,   262044,    69264,     5040;\n  8: 1,  989,  60344,  803150,  3415900,  5355844,  3173256,   663696,   40320;\n  9: 1, 2009, 202784, 4317260, 29890000, 80405884, 89624136, 40897656, 6999840, 362880;\n","recurrence_sentence":"The row polynomials satisfy (n-2+x)*P(n,x) = -((4*n^3-20*n^2+31*n-15)*x^2+(6*n^2-21*n+18)*x^3+(2*n-3)*x^4)*P(n-2,x) - ((2*n^2-6*n+4)*x^2-(2*n^2-9*n+9)*x^3-(3*n-6)*x^4-x^5)*d/dx P(n-2,x) + (n-2+(4*n^2-13*n+11)*x+(5*n-8)*x^2+x^3)*P(n-1,x) + ((n-2)*x-(n-3)*x^2-x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[(-((4*n^3 - 20*n^2 + 31*n - 15)*x^2 + (6*n^2 - 21*n + 18)*x^3 + (2*n - 3)*x^4)*P[n - 2, x] - ((2*n^2 - 6*n + 4)*x^2 - (2*n^2 - 9*n + 9)*x^3 - (3*n - 6)*x^4 - x^5)*D[P[n - 2, x], x] + (n - 2 + (4*n^2 - 13*n + 11)*x + (5*n - 8)*x^2 + x^3)*P[n - 1, x] + ((n - 2)*x - (n - 3)*x^2 - x^3)*D[P[n - 1, x], x])/(n - 2 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (-((4*n^3 - 20*n^2 + 31*n - 15)*x^2 + (6*n^2 - 21*n + 18)*x^3 + (2*n - 3)*x^4)*v[i - 2] - ((2*n^2 - 6*n + 4)*x^2 - (2*n^2 - 9*n + 9)*x^3 - (3*n - 6)*x^4 - x^5)*deriv(v[i - 2], x) + (n - 2 + (4*n^2 - 13*n + 11)*x + (5*n - 8)*x^2 + x^3)*v[i - 1] + ((n - 2)*x - (n - 3)*x^2 - x^3)*deriv(v[i - 1], x))/(n - 2 + x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e14e37227fef094e","steps":["reverse","D+1"],"edges":["e-c8375d3b3fe52ed3","e-ef8e5aa6bfa454a9"],"text":"A163936 \u2192 reverse \u2192 D+1"},"headline":"A163936 \u2192 reverse \u2192 D+1","chain":null}